Jeffrey S. Rosenthal (@ProbabilityProf) is a Professor of Statistics at the University of Toronto, and the author of the book Struck by Lightning: The Curious World of Probabilities.

Last week, Ontario voters elected Premier Doug Ford. Or did they? In fact, nearly 60% of Ontario voters would have preferred another premier. So how did the Conservatives win a comfortable majority of the seats?

The answer is, through vote splitting. Those 60% of voters partitioned their votes among three broadly center-left parties: the NDP, the Liberals, and the Greens. In our riding-by-riding "first-past-the-post" plurality voting system, this allowed the Conservatives to win a majority of the seats with just 40% of the votes, to the immense frustration of many.

As a center-left voter who also happens to be a statistician, I decided to investigate further. How much did vote splitting affect the results? If center-left parties had, hypothetically speaking, been able to combine their votes together, would they have been able to win more seats?

The answer is a resounding yes. The NDP won 40 seats, the Liberals 7, and the Greens 1, for a total of 48 seats, far less than half of the 124 up for grabs. But by my calculations, if the NDP and Liberal and Green votes were all combined together, this would have instead led them to a massive majority of 90 seats, leaving just 34 for the Conservatives.

Or, if just the NDP and Liberal votes were combined together, this would still have led to a comfortable majority of 82 seats -- much more than the NDP and Liberal combined total of 47 seats in the actual election.

I was especially curious about vote splitting with the Greens. I have several friends who have been candidates for the Green Party, attracted to its fresh approach and environmental priorities. But I always suspected that their effort would further split the center-left vote, and thus primarily benefit the Conservatives. Did it?

Well, if the NDP and Green (but not Liberal) votes were combined together, they would have won a total of 50 seats instead of their actual 41, taking an additional eight from the Conservatives plus one from the Liberals. Would that have been enough to win the election? No.

But if the NDP and Greens had then increased their combined votes by just a further 18% in each riding, they would have formed a majority government. By contrast, the NDP vote alone would have had to increase by a whopping 42% in each riding to achieve the same victory.

And, if the Green Party continues to grow -- buoyed by leader Mike Schreiner's historic victory in Guelph -- then we might soon see an election in which the Conservatives win a majority even though combining *any* two of the center-left parties' votes together would have been enough to ensure victory. The more center-left parties, and the more equal their vote totals, the better the Conservatives chances become.

A similar pattern has emerged at the federal level. In the 2011 Canadian election, the center-left vote was split sufficiently evenly among the NDP, Liberals, Bloc Quebecois, and Greens to give Stephen Harper's Conservatives a majority of the seats despite less than 40% of the votes. But in 2015, Justin Trudeau's Liberals received over twice as many votes as the NDP, thus breaking the NDP-Liberal deadlock, giving the Liberals a majority of the seats, and relegating the Conservatives to the opposition benches.

Last week's election initially looked like a repeat of 2015. The governing Liberal vote was collapsing, and it seemed that progressive voters might all move to the NDP and propel them to victory. However, Kathleen Wynne's strong late campaigning and dramatic pre-election concession speech allowed the Liberals to hang on to just enough of their support to continue the old vote splitting patterns.

As for Ontario's future, the statistics are clear. If the center-left parties continue to split their votes fairly evenly, then the Conservatives will continue to win the most seats. But if center-left voters manage to coalesce around any one single party, then their victories will be assured for years to come.

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